# Chapter 4. Supersymmetric Quantum Mechanics on the 2-sphere

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 4 Supersymmetric Quantum Mechanics on the 2-sphere Appendix 4(i) Complex Manifolds [12], [13] A complex manifold M is a differentiable manifold which can be covered by a set of complex coordinate patches, and where the coordinate transformations on the overlap between the patches are holomorphic On a complex manifold the cotangent space decomposes into the sum of a holomorphic and an anti-holomorphic cotangent space, so p-forms may be treated as (q, r)-forms, which q and r are the holomorphic and antiholomorphic degrees respectively, and p = q + r The exterior derivative on a complex manifold splits naturally into where, the holomorphic derivative map (q, r)-forms to (q+1, r)-forms and, the anti-holomorphic derivative maps (q, r)-forms to (q, r+1)-forms The nilpotence of the exterior derivative implies the relations The Hodge Star can now be used to define the adjoint operators, so that in general on a complex manifold there are three different Laplacians:,, On a complex manifold a Hermitean metric may be defined where is a positive definite Hermitean matrix for each z The associated (1, 1)-form

2 is called the Kähler form A manifold with closed Kähler form,, is known as a Kähler manifold An important property of Kähler manifolds is that the three different Laplacians defined on them are equal up to a constant This relation leads to Hodge duality where are the Hodge Numbers of M, the dimension of harmonic (p, q)-forms In terms of supersymmetry the fact that the different Laplacians are equal means that the two supersymmetry operators can be split into four,,, which satisfy the supersymmetry algebra ;,, so that the non-zero energy solutions form quadruplets under supersymmetry rather than doublets, while the zero energy solutions are still singlets Poincare duality doubles the degeneracy of excited states to eightfold This does not, however, affect the analysis of section 31 where a Killing Vector was introduced into the Hamiltonian, as a Killing Vector is necessarily real so the degeneracy due to supersymmetry is the same whether or not the manifold is Kähler On a compact complex manifold of n complex dimensions the volume form V is proportional to the n-th power exterior product of the Kähler form If the manifold is Kähler this relationship along with the Kähler condition,, implies that is a representative of the cohomology of the manifold for all k, The Kähler condition implies is closed, but if it was exact,, by Stokes Theorem, the volume would have to be zero In virtually all of the following, complex coordinates will be used, as this results in considerable simplifications The functions which are defined will usually be written with arguments which are holomorphic, this is an abbreviation, ie None of the functions are actually holomorphic

3 4 Supersymmetric Quantum Mechanics on the 2-sphere The simplest non-trivial illustrations of Witten s ideas about fixed point theorems and supersymmetric quantum mechanics is on the 2-sphere In this section the supersymmetric Schrödinger Equation on the 2- sphere is solved; the solutions form multiplets under the isometry group SU(2) An infinitesimal generator of an isometry, a Killing Vector, is then introduced into the supersymmetry algebra as in section 3 This breaks the symmetry under the isometry group and leads to a new Hamiltonian which cannot be solved exactly for excited states, though zero energy solutions can be found The zero energy solutions are related to the topology of the 2-sphere in the way expect from Witten s paper [1] Perturbation theory corrections to the spectrum of the excited states are calculated The asymptotic regime is also examined and related to Witten s analysis Finally, the various approximations to the energy level are compared to those found numerically by a computer program 41 The 2-sphere As a complex manifold the 2-sphere is equivalent to It can therefore be defined in terms of two homogeneous complex coordinates and with points identified which are equal up to a scaling by a complex number The 2-sphere can be covered with two patches by using the inhomogeneous coordinates everywhere except everywhere except A metric may be put on the 2-sphere by stereographic projection In terms of the coordinates z, this is The group of metric preserving rotations in two flat complex dimensions is U(2) The infinitesimal generators of this group may be spanned by the operators,,, The isometry group of the 2-sphere is SU(2) The generators of this group may be obtained from the generators of U(2) by using the chain rule to rewrite them in terms of the inhomogeneous coordinate z

4 These three operators, and the two ladder operators, form a basis for SU(2) The operator is equal to and so doesn t give anything new The freedom to scale the homogeneous coordinates on the sphere by a complex number has removed a U(1) factor from the isometry group The commutation relations of the SU(2) generators are,, Rewriting in terms of and gives and the commutation relations run over 411 The Ordinary Laplacian As on the 2-torus the 4x4 matrix notation for operators will be used Wavefunctions on the sphere will be of the form and will be denoted by the 4x1 column matrix Using this notation the holomorphic and anti-holomorphic derivatives may be written as

5 and by applying the Hodge Star Map their adjoints may be found: where means that acts on the product of and the wavefunction Because the 2-sphere is a Kähler manifold there are four supersymmetry operators as opposed to two in the generic case These supersymmetry operators are

6 and, The Hamiltonian is the square of these operators, the Laplacian The zero energy eigensolutions correspond to the cohomology of the 2-sphere; namely the constant zero-form and the volume form The excited states may be found by considering the Hamiltonian on zero-forms and then using the supersymmetry operators to find the one-form and the two-form solutions On zero-forms the Hamiltonian is simply The isometries of the 2-sphere are rotations of the manifold which preserve the metric, thus the generators of SU(2) commute with the Laplacian and so its eigensolutions form representations os SU(2) The highest weight solutions are of the form

7 which can easily be shown to satisfy the Schrödinger Equation with energy These solutions have highest weight because they are annihilated by the raising operator and the eigenvalue of the operator is n The rest of the solutions in the multiplets can be found by applying the lowering operator, which lowers the eigenvalue of by one at each step These multiplets have energy and each member is labelled by the eigenvalue of, the angular momentum around the equator of the sphere, which takes the values These solutions equal the Associated Legendre Functions, where For example, the first excited states form the adjoint representation of SU(2) and have energy E = 2 Having solved the Schrödinger Equation for (0,0)-forms, we are now in the position of being able to find the (1,0)-form, (0,1)-form and (1,1)-form solutions Acting with the supersymmetry operator on a (0,0)-form solution gives a (1,0)-form solution

8 and acting with gives a (0,1)-form solution while acting consecutively with both supersymmetry operators will give a (1,1)-form solution so supersymmetry produces a four-fold degeneracy on top of the (2n+1)-fold degeneracy due to SU(2) symmetry

9 42 Inclusion of a Killing Vector into the Supersymmetry Algebra Following Witten we will now generalize the exterior derivative by the addition of the operation of taking the interior product with a Killing Vector k where s is a positive parameter This leads to a supersymmetry algebra of the form of equations (1) of section 34 On the 2-sphere the simplest Killing Vector to use is the generator of rotations around the equator The factor of i is necessary because is imaginary and a Killing Vector must be real Writing out the operators and explicitly using the Killing Vector gives Defining the supersymmetry operators as, leads to the Hamiltonian (1) where H is the ordinary Laplacian and is the 4x4 unit matrix The Hamiltonian operator has been defined such that This is because the holomorphic Laplacian, on a Kähler manifold, equals half the ordinary Laplacian, see Appendix 4(i) The new s-dependent terms correspond to the terms in section 3 multiplied by a half

10 where and is its dual, The central charge P takes the form The Schrödinger Equation can be diagonalized on even-forms, ie (0,0)-forms and (1,1)-forms by writing the solutions in terms of the self and anti-self-dual combinations of the unit (0,0)-form and the Kähler form These solutions are self and anti-self-dual respectively under the operation of i times the Hodge Star The factor of i comes from the alteration in the definition of the Hodge Star when the exterior algebra is taken to be complex rather than real This is due to the fact that if, Acting with the Hamiltonian on gives for the (0,0)-form piece, the equation (2) The equation for the (1,1)-form piece is exactly the same So solving the Schrödinger Equation on even-forms reduces to finding solutions of this equation The odd-form solutions can then be found by applying the supersymmetry operators The excited states have an eight-fold degeneracy; and are contained in separate supersymmetry quadruplets related by Poincare duality The other members of the quadruplets may be found by acting with the supersymmetry operators

11 so the quadruplet contains an anti-self-dual even-form as well as a self-dual even-form The other quadruplet which contains is obtained from this one by exchanging and As s tends to zero becomes equal to and the quadruplets converge to become identical It is now obvious what the zero energy eigensolutions of are as these must be annihilated by and For this to occur either must be real, satisfying and the complex conjugate equation or must be real and satisfy Thus there are two zero energy eigensolutions which is self-dual and

12 which is anti-self-dual As s tends to zero these solutions tend to the sum and difference of the representatives of the cohomology of the 2-sphere The fact that the zero energy solutions still exist means that supersymmetry is unbroken; unlike the case of the 2-torus, see Appendix 3(iii), where the introduction of a Killing Vector removes all zero energy states The zero energy solutions are the only ones that can be found exactly To study the spectrum of the excited states approximate methods must be used The equations for may be further simplified by using the zero energy solutions as integrating factors Substituting into equation (412) gives immediately (3) where the Schrödinger Equation now takes the form of the Laplacian plus a term proportional to the operator This vector commutes with the Killing Vector, but not with the other two generators of SU(2), so m, the eigenvalue of, is still a good quantum number, but the full SU(2) symmetry has been broken to U(1) There are two sets of excited states which correspond to excitations around the two zero energy ground states The signs of the parameter s in equation (3) for these two sets of states are opposite, however it turns out that the energy levels are the same in both cases

13 43 Perturbation Theory For small s the term in the Schrödinger Equation may be treated as a perturbation of Legendre s Equation This vector can be rewritten so that its effect on Associated Legendre Functions is much more obvious are the Associated Legendre Functions with and are the SU(2) raising and lowering operators The effect of this operator is therefore to conserve m and change n by plus or minus one This indicates that the first order shift in energy is zero for all n and m In terms of the coordinates on the 2-sphere the vanishing of that the perturbation is anti-symmetric under the change of coordinates is due to the fact, corresponding to the inversion of the sphere This is due to the opposite direction of the rotation around the sphere s equation when viewed from the other pole Applying this change of coordinates to the matrix elements gives which after taking the complex conjugate of the right hand side implies that the matrix elements must vanish Thus all odd orders of perturbation theory are zero, so the sign of s is irrelevant and the excitations around the two different ground states are degenerate to all orders in perturbation theory Each excited state is a member of a quadruplet, so the total degeneracy due to supersymmetry is eight-fold

14 431 Second Order Perturbation Theory In the formula for the second order perturbation theory shift in energy which is in principle an infinite sum Only the two terms, with contribute The easiest way to calculate the shift is to use a formula from the theory of Legendre Functions, where In terms of these coordinates, where and are the longitude and latitude on the sphere, the perturbation takes the form The effect of this operator on the Associated Legendre Functions is therefore (1) Substituting this formula into the expression for the second order energy shift gives

15 This formula depends on, so the SU(2) (2n+1)-plets are split into n doublets and a singlet On the lowest of the excited representations, the 3 which has energy and the 5 which has energy, the second order energy changes due to the perturbation are 432 Fourth Order Perturbation Theory The next non-zero order of perturbation theory is fourth order The corrections to the energy to this order may be found by a substitution of the form of into equation (3) of section 42 and use of formula (1) of section 431 The energy to this order is Equating coefficients of, and in this equation, in the case where, leads to the expression Substituting in the value, for the first excited state with m = 0, gives the fourth order energy shift, The next case n = 2, gives

16 44 Asymptotic Solutions For large s the spectrum of the Schrödinger Equation can be found as an asymptotic expansion in The low energy solutions are localized around the fixed points of the vector, which are where the Hamiltonian is smallest These fixed points are the same as those of, and, the south and north poles of the sphere 441 The Harmonic Oscillator Approximation In the large s limit the Hamiltonian tends to a supersymmetric harmonic oscillator around these fixed points Substituting into equation (1) of section 42 we obtain Taking s large and expanding in powers of gives the approximate form of the Schrödinger Equation near the south pole Neglecting terms of order and smaller and substituting back into the equation gives the harmonic oscillator equation

17 The coordinate z is not well defined at the other fixed point of, so the second coordinate patch must be utilised Using the coordinate, the Schrödinger Equation becomes where a factor of has been inserted into the off-diagonal matrix to compensate for the transformation of the two-form For small w, near the north pole, this equation takes the same form as that near the south pole apart from the alteration, due to the opposite manner of the rotation around the equator when the sphere is viewed from the opposite pole To write the Killing Vector in the standard form, where it is multiplied by a positive parameter, interchanges the coordinates and, so that the coordinates in the second patch have opposite orientation to those in the first This explains why the zero energy solutions near the two poles have opposite duality The zero energy solution near the south pole is self-dual whilst that near the north pole is anti-self-dual The eight-fold degeneracy of the excited states is now realised as a degeneracy between two quadruplets, one at each pole The general form of the excited states can be found using harmonic oscillator ladder operators At the sound pole the solutions are

18 with energy and with the eigenvalue of the Lie Derivative along the Killing Vector, As these states are harmonic oscillator eigenstates, they form SU(2) representations of dimension where When the corrections to the energy are included this SU(2) symmetry is broken to U(1) To this approximation the supersymmetry operators act on the even-form solutions with harmonic oscillator lowering operators to give the odd-form solutions as with energy and respectively Both sets of states have a (p+q)-fold degeneracy The other even-form states in the supersymmetry multiplets, obtained by acting on with both and consecutively are where and and the degeneracy is - fold The total degeneracy of even-form solutions of energy localized around the south pole, as for oddforms, is The results near the north pole are completely analogous

19 442 Asymptotic Corrections The second order terms in the asymptotic expansion of the energy are of order one and so don t vanish in the large s limit The easiest way to calculate these corrections to the harmonic oscillator energies is to use the form of the Schrödinger Equation, equation (3) of section 42 The plus sign is taken corresponding to excitations around the self-dual ground state, which is localized around the south pole for large 2 When considering solutions localized around the north pole the sign in the equation is again positive because the minus sign corresponding to the anti-self-dual ground state is cancelled by the fact that when using the coordinate around the north pole Because of this the corrections are the same at both poles Putting and keeping terms up to gives the equation Inserting as an expansion in powers of and gives which implies the following equation for the coefficients in the solutions For a closed series solution, which gives the energy to as The - fold degeneracy of the harmonic oscillator has been split, so that states either form doublets, produced by interchanging p and q in the formula for, or singlets if This is due to the U(1) symmetry of There is of course still the eight-fold degeneracy due to supersymmetry as well The oddform solutions have energies and and the other even-form solutions have The spectrum of the low lying states in this approximation and the degeneracy of the even-form states near the south pole is the following

20

21 45 Summary After the inclusion of the Killing Vector the asymptotic zero energy solutions localized one near each fixed point, are in a one-to-one correspondence with the exact zero energy solutions Near the south pole and near the north pole This correspondence is due to Witten s relation By the index arguments of section 3, the number of even-form solutions minus the number of odd-form solutions must be independent of s For on the 2-sphere, the zero energy solutions corresponding to the representatives of the cohomology are the constant (0,0)-form and the Kähler form, both of which are even-forms, therefore For non-zero s there are again two even-form zero energy solutions and no odd-form ones and ultimately in the large s limit these solutions become localized one at each fixed point This corresponds to the Lefschetz Fixed Point Theorem, number of fixed point The excited states, which form the (2n+1)-dimensional representations of SU(2) when s = 0 are more difficult to analyse for non-zero s, but approximate methods show how the SU(2) symmetry breaks to U(1)

22 Appendix 4(ii) Computer Analysis After the introduction of the Killing Vector into the supersymmetry algebra, the Schrödinger Equation (3) of section 42 is not soluble analytically for excited states, so a computer program was written to find accurate values of the eigenvalues of this equation for all values of s The program uses the library subroutine D02KDF from the NAG FORTRAN library, which finds a specified eigenvalue E of a Sturm- Liouville system defined by a second order self-adjoint differential equation, together with boundary conditions at the endpoints a and b To transform equation (413) into a suitable form for the computer analysis the substitution where, was used leading to the equation, which is the Associated Legendre s Equation plus the s-dependent term The operator is equivalent to the central charge P and commutes with the Hamiltonian, so the solutions of the Schrödinger Equation may be taken to be eigensolutions of this operator with eigenvalue m, thus reducing the equation to an ordinary differential equation The self-adjoint form necessary for specification in the library subroutine is given by so that the functions and are The boundary conditions are specified in YL(1) and YL(2) by the values of and at the other end point, only the ratios of the two values being important This is the program used to find the eigenvalues of the Schrödinger Equation on the central charge PROGRAM SUSYS2 REAL *8 DELAM, ELAM, TOL, PI, D, HMAX(2, 4), XPOINT(4), M, S INTEGER IFAIL, K, MAXIT EXTERNAL BDYVL, COEFF, MONIT after the introduction of

23 COMMON PI, D, M, S PRINT *, ENTER M, S, K, ELAM READ *, M, S, K, ELAM D = 0001 TOL = DELAM = 01 PI = X01AAF(D) MAXIT = 0 XPOINT(1) = 00 XPOINT(2) = 0001 XPOINT(3) = PI-0001 XPOINT(4) = PI HMAX(1, 1) = 00 IFAIL = 0 CALL D02KDF (XPOINT, 4, COEFF, BDYVL, K, TOL, ELAM, DELAM, HMAX, MAXIT, 0, MONIT, IFAIL) WRITE (*, *) K, ELAM, DELAM, IFAIL STOP END SUBROUTINE MONIT (MAXIT, IFLAG, ELAM, FINFO) INTEGER MAXIT, IFLAG, I REAL *8 ELAM, FINFO (15), PI, D, M, S COMMON PI, D, M, S WRITE (*, *) MAXIT, IFLAG, ELAM, (FINFO(I), I = 1, 4) RETURN END SUBROUTINE BDYVL (XL, XR, ELAM, YL, YR) REAL *8 ELAM, XL, XR, PI, D, YL(3), YR(3), M, S COMMON PI, D, M, S YL(1) = 10 YL(2) = M*EXP(S) YR(1) = 10 YR(2) = -M*EXP(-S) RETURN END SUBROUTINE COEFF (P, Q, DQDL, X, ELAM, JINT) REAL *8 P, Q, M, S, DQDL, X, ELAM, PI, D INTEGER JINT

24 COMMON PI, D, M, S P = -SIN(X)*EXP(S*COS(X)) Q = -P*(M*M/(SIN(X)*SIN(X)) ELAM) DQDL = P RETURN END The numerical results from the computer show how the energies of the eigenstates as functions of the parameter s interpolates between the perturbative regime and the asymptotic regime, and verify the approximate analysis in these two cases The first two graphs compare the results from the computer analysis with second order and fourth order perturbation theory and also with the asymptotic results in the cases of the first two excited states with These graphs show a good agreement between the perturbative results and the numerical values from the computer for surprisingly large values of s The third graph shows how the spectrum varies as s increases for the low lying energy levels The values of on the right-hand side of the graph indicate the level of the corresponding Legendre Funciton which the eigensolution tends to as s tends to zero The graph shows the symmetry breaking which splits the SU(2) -plets into doublets and a singlet Asymptotically the graphs have gradients proportional to s multiplied by an integer corresponding to the harmonic oscillator energies, with a degeneracy of Taking into account the corrections the graphs show agreement with the energy levels given at the end of section 4 Graph 3, for, in fact only shows half the even-form solutions Apart from the zero energy solutions each even-form solution is paired with an odd-form solution

25 Graph 1 on the following page shows the energy of the first excited state with m = 0 for different values of the parameter s calculated using the various approximate methods The crosses mark the results of the computer analysis

26

27 Graph 2 on the following page shows the energy of the second excited state with m = 0 for different values of the parameter s calculated using the various approximate methods The crosses mark the results of the computer analysis

28

29 Graph 3 on the following page shows the lowest energy levels as functions of s as given by the computer analysis When s = 0 at the left-hand side the solutions are Associated Legendre Functions, on the right the energies tend to the asymptotic vales, is the level of the harmonic oscillator solution

30

### Particle Physics. Michaelmas Term 2011 Prof Mark Thomson. Handout 7 : Symmetries and the Quark Model. Introduction/Aims

Particle Physics Michaelmas Term 2011 Prof Mark Thomson Handout 7 : Symmetries and the Quark Model Prof. M.A. Thomson Michaelmas 2011 206 Introduction/Aims Symmetries play a central role in particle physics;

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### CHAPTER 12 MOLECULAR SYMMETRY

CHAPTER 12 MOLECULAR SYMMETRY In many cases, the symmetry of a molecule provides a great deal of information about its quantum states, even without a detailed solution of the Schrödinger equation. A geometrical

### 1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as

Chapter 3 (Lecture 4-5) Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series

### Rutgers - Physics Graduate Qualifying Exam Quantum Mechanics: September 1, 2006

Rutgers - Physics Graduate Qualifying Exam Quantum Mechanics: September 1, 2006 QA J is an angular momentum vector with components J x, J y, J z. A quantum mechanical state is an eigenfunction of J 2 J

### H = = + H (2) And thus these elements are zero. Now we can try to do the same for time reversal. Remember the

1 INTRODUCTION 1 1. Introduction In the discussion of random matrix theory and information theory, we basically explained the statistical aspect of ensembles of random matrices, The minimal information

### Quantum Mechanics and Representation Theory

Quantum Mechanics and Representation Theory Peter Woit Columbia University Texas Tech, November 21 2013 Peter Woit (Columbia University) Quantum Mechanics and Representation Theory November 2013 1 / 30

### The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23

(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, high-dimensional

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### Chapter 9 Unitary Groups and SU(N)

Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Classical Physics Prof. V. Balakrishnan Department of Physics Indian Institution of Technology, Madras. Lecture No. # 13

Classical Physics Prof. V. Balakrishnan Department of Physics Indian Institution of Technology, Madras Lecture No. # 13 Now, let me formalize the idea of symmetry, what I mean by symmetry, what we mean

### We consider a hydrogen atom in the ground state in the uniform electric field

Lecture 13 Page 1 Lectures 13-14 Hydrogen atom in electric field. Quadratic Stark effect. Atomic polarizability. Emission and Absorption of Electromagnetic Radiation by Atoms Transition probabilities and

### Hermitian Operators An important property of operators is suggested by considering the Hamiltonian for the particle in a box: d 2 dx 2 (1)

CHAPTER 4 PRINCIPLES OF QUANTUM MECHANICS In this Chapter we will continue to develop the mathematical formalism of quantum mechanics, using heuristic arguments as necessary. This will lead to a system

### From Fourier Series to Fourier Integral

From Fourier Series to Fourier Integral Fourier series for periodic functions Consider the space of doubly differentiable functions of one variable x defined within the interval x [ L/2, L/2]. In this

### State of Stress at Point

State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

### The Quantum Harmonic Oscillator Stephen Webb

The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems

### Advanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Advanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. # 28 Fourier Series (Contd.) Welcome back to the lecture on Fourier

### Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds

Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative

### Entangling rates and the quantum holographic butterfly

Entangling rates and the quantum holographic butterfly David Berenstein DAMTP/UCSB. C. Asplund, D.B. arxiv:1503.04857 Work in progress with C. Asplund, A. Garcia Garcia Questions If a black hole is in

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### Differential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation

Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of

### WORK SCHEDULE: MATHEMATICS 2007

, K WORK SCHEDULE: MATHEMATICS 00 GRADE MODULE TERM... LO NUMBERS, OPERATIONS AND RELATIONSHIPS able to recognise, represent numbers and their relationships, and to count, estimate, calculate and check

### Sequences and Series

Contents 6 Sequences and Series 6. Sequences and Series 6. Infinite Series 3 6.3 The Binomial Series 6 6.4 Power Series 3 6.5 Maclaurin and Taylor Series 40 Learning outcomes In this Workbook you will

### T Duality in M(atrix) Theory and S Duality in field theory

SU-ITP-96-12 April 1996 hep-th/9611164 T Duality in M(atrix) Theory and S Duality in field theory Leonard Susskind Department of Physics Stanford University, Stanford, CA 94305-4060 The matrix model formulation

### Solving a System of Equations

11 Solving a System of Equations 11-1 Introduction The previous chapter has shown how to solve an algebraic equation with one variable. However, sometimes there is more than one unknown that must be determined

### Summary of week 8 (Lectures 22, 23 and 24)

WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry

### SECTION 1.5: PIECEWISE-DEFINED FUNCTIONS; LIMITS AND CONTINUITY IN CALCULUS

(Section.5: Piecewise-Defined Functions; Limits and Continuity in Calculus).5. SECTION.5: PIECEWISE-DEFINED FUNCTIONS; LIMITS AND CONTINUITY IN CALCULUS LEARNING OBJECTIVES Know how to evaluate and graph

### For the case of an N-dimensional spinor the vector α is associated to the onedimensional . N

1 CHAPTER 1 Review of basic Quantum Mechanics concepts Introduction. Hermitian operators. Physical meaning of the eigenvectors and eigenvalues of Hermitian operators. Representations and their use. on-hermitian

### The Characteristic Polynomial

Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

### Exam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form.

Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? To solve an equation is to find the solution set, that is, to find the set of all elements in the domain of the

### Zeros of Polynomial Functions

Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

### Topic 1: Matrices and Systems of Linear Equations.

Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method

### Lecture No. # 02 Prologue-Part 2

Advanced Matrix Theory and Linear Algebra for Engineers Prof. R.Vittal Rao Center for Electronics Design and Technology Indian Institute of Science, Bangalore Lecture No. # 02 Prologue-Part 2 In the last

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

### Metrics on SO(3) and Inverse Kinematics

Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction

### 5 Homogeneous systems

5 Homogeneous systems Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m

### A. The wavefunction itself Ψ is represented as a so-called 'ket' Ψ>.

Quantum Mechanical Operators and Commutation C I. Bra-Ket Notation It is conventional to represent integrals that occur in quantum mechanics in a notation that is independent of the number of coordinates

### Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010

Section 4.1: Systems of Equations Systems of equations A system of equations consists of two or more equations involving two or more variables { ax + by = c dx + ey = f A solution of such a system is an

### Tangent and normal lines to conics

4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

### Introduction to Matrix Algebra I

Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model

### Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

### Mixed states and pure states

Mixed states and pure states (Dated: April 9, 2009) These are brief notes on the abstract formalism of quantum mechanics. They will introduce the concepts of pure and mixed quantum states. Some statements

### Chapter 1 - Matrices & Determinants

Chapter 1 - Matrices & Determinants Arthur Cayley (August 16, 1821 - January 26, 1895) was a British Mathematician and Founder of the Modern British School of Pure Mathematics. As a child, Cayley enjoyed

### Quick Reference Guide to Linear Algebra in Quantum Mechanics

Quick Reference Guide to Linear Algebra in Quantum Mechanics Scott N. Walck September 2, 2014 Contents 1 Complex Numbers 2 1.1 Introduction............................ 2 1.2 Real Numbers...........................

### a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

### 6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)

6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,

### 3. A LITTLE ABOUT GROUP THEORY

3. A LITTLE ABOUT GROUP THEORY 3.1 Preliminaries It is an apparent fact that nature exhibits many symmetries, both exact and approximate. A symmetry is an invariance property of a system under a set of

### LS.6 Solution Matrices

LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

### PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Test 3 November 22, 2004

PHY464 Introduction to Quantum Mechanics Fall 4 Practice Test 3 November, 4 These problems are similar but not identical to the actual test. One or two parts will actually show up.. Short answer. (a) Recall

### Factoring Polynomials

UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### The Solution of Linear Simultaneous Equations

Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve

### The Central Equation

The Central Equation Mervyn Roy May 6, 015 1 Derivation of the central equation The single particle Schrödinger equation is, ( H E nk ψ nk = 0, ) ( + v s(r) E nk ψ nk = 0. (1) We can solve Eq. (1) at given

### The Mathematics of Origami

The Mathematics of Origami Sheri Yin June 3, 2009 1 Contents 1 Introduction 3 2 Some Basics in Abstract Algebra 4 2.1 Groups................................. 4 2.2 Ring..................................

### ALGEBRA I A PLUS COURSE OUTLINE

ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best

### Synchronization in Electrical Power Network. Oren Lee, Thomas Taylor, Tianye Chi, Austin Gubler, Maha Alsairafi

Synchronization in Electrical Power Network Oren Lee, Thomas Taylor, Tianye Chi, Austin Gubler, Maha Alsairafi Contents Introduction 1 Synchronization 1 Enhancement of synchronization stability 3 Future

### 1 Symmetries of regular polyhedra

1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an

### Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

### Chapter 4 - Systems of Equations and Inequalities

Math 233 - Spring 2009 Chapter 4 - Systems of Equations and Inequalities 4.1 Solving Systems of equations in Two Variables Definition 1. A system of linear equations is two or more linear equations to

### Min-cost flow problems and network simplex algorithm

Min-cost flow problems and network simplex algorithm The particular structure of some LP problems can be sometimes used for the design of solution techniques more efficient than the simplex algorithm.

### Mathematical Procedures

CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

### PSS 27.2 The Electric Field of a Continuous Distribution of Charge

Chapter 27 Solutions PSS 27.2 The Electric Field of a Continuous Distribution of Charge Description: Knight Problem-Solving Strategy 27.2 The Electric Field of a Continuous Distribution of Charge is illustrated.

### Contents. Gbur, Gregory J. Mathematical methods for optical physics and engineering digitalisiert durch: IDS Basel Bern

Preface page xv 1 Vector algebra 1 1.1 Preliminaries 1 1.2 Coordinate System invariance 4 1.3 Vector multiplication 9 1.4 Useful products of vectors 12 1.5 Linear vector Spaces 13 1.6 Focus: periodic media

### 8.1 Examples, definitions, and basic properties

8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A k-form ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)-form σ Ω k 1 (M) such that dσ = ω.

### Math Rational Functions

Rational Functions Math 3 Rational Functions A rational function is the algebraic equivalent of a rational number. Recall that a rational number is one that can be epressed as a ratio of integers: p/q.

### 1 Gaussian Elimination

Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 Gauss-Jordan reduction and the Reduced

### Fourier Analysis. A cosine wave is just a sine wave shifted in phase by 90 o (φ=90 o ). degrees

Fourier Analysis Fourier analysis follows from Fourier s theorem, which states that every function can be completely expressed as a sum of sines and cosines of various amplitudes and frequencies. This

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field

3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 77 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field Overview: The antiderivative in one variable calculus is an important

### 4. The Infinite Square Well

4. The Infinite Square Well Copyright c 215 216, Daniel V. Schroeder In the previous lesson I emphasized the free particle, for which V (x) =, because its energy eigenfunctions are so simple: they re the

### Lesson 17: Graphing the Logarithm Function

Lesson 17 Name Date Lesson 17: Graphing the Logarithm Function Exit Ticket Graph the function () = log () without using a calculator, and identify its key features. Lesson 17: Graphing the Logarithm Function

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

### Name: Section Registered In:

Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

### (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

### The Essentials of Quantum Mechanics

The Essentials of Quantum Mechanics Prof. Mark Alford v7, 2008-Oct-22 In classical mechanics, a particle has an exact, sharply defined position and an exact, sharply defined momentum at all times. Quantum

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### Trigonometry Notes Sarah Brewer Alabama School of Math and Science. Last Updated: 25 November 2011

Trigonometry Notes Sarah Brewer Alabama School of Math and Science Last Updated: 25 November 2011 6 Basic Trig Functions Defined as ratios of sides of a right triangle in relation to one of the acute angles

### Simple harmonic motion

PH-122- Dynamics Page 1 Simple harmonic motion 02 February 2011 10:10 Force opposes the displacement in A We assume the spring is linear k is the spring constant. Sometimes called stiffness constant Newton's

### with "a", "b" and "c" representing real numbers, and "a" is not equal to zero.

3.1 SOLVING QUADRATIC EQUATIONS: * A QUADRATIC is a polynomial whose highest exponent is. * The "standard form" of a quadratic equation is: ax + bx + c = 0 with "a", "b" and "c" representing real numbers,

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### 5.61 Physical Chemistry 25 Helium Atom page 1 HELIUM ATOM

5.6 Physical Chemistry 5 Helium Atom page HELIUM ATOM Now that we have treated the Hydrogen like atoms in some detail, we now proceed to discuss the next simplest system: the Helium atom. In this situation,

### Mathematical Physics, Lecture 9

Mathematical Physics, Lecture 9 Hoshang Heydari Fysikum April 25, 2012 Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 1 / 42 Table of contents 1 Differentiable manifolds 2 Differential

### 6 J - vector electric current density (A/m2 )

Determination of Antenna Radiation Fields Using Potential Functions Sources of Antenna Radiation Fields 6 J - vector electric current density (A/m2 ) M - vector magnetic current density (V/m 2 ) Some problems

### Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013

Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,

### Let s first see how precession works in quantitative detail. The system is illustrated below: ...

lecture 20 Topics: Precession of tops Nutation Vectors in the body frame The free symmetric top in the body frame Euler s equations The free symmetric top ala Euler s The tennis racket theorem As you know,

### Year 1 Maths Expectations

Times Tables I can count in 2 s, 5 s and 10 s from zero. Year 1 Maths Expectations Addition I know my number facts to 20. I can add in tens and ones using a structured number line. Subtraction I know all

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course

### THE SIGN OF A PERMUTATION

THE SIGN OF A PERMUTATION KEITH CONRAD 1. Introduction Throughout this discussion, n 2. Any cycle in S n is a product of transpositions: the identity (1) is (12)(12), and a k-cycle with k 2 can be written

### Integrals of Rational Functions

Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t

### Orthogonal Diagonalization of Symmetric Matrices

MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding

### Basic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics. Indian Institute of Technology, Delhi

Basic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics. Indian Institute of Technology, Delhi Module No. # 02 Simple Solutions of the 1 Dimensional Schrodinger Equation Lecture No. # 7. The Free

### Definition 12 An alternating bilinear form on a vector space V is a map B : V V F such that

4 Exterior algebra 4.1 Lines and 2-vectors The time has come now to develop some new linear algebra in order to handle the space of lines in a projective space P (V ). In the projective plane we have seen